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Question : 27
Total: 29
Show that the height of a closed right circular cylinder of given surface and maximum volume, is equal to the diameter of its base.
Solution:
Let r and h be the radius and height of the cylinder. Then,
A = 2πrh +2 π r 2
⇒ h =
Now, Volume (V) =Ï€ r 2 h
⇒ V =π r 2 (
)
( A r − 2 π r 3 )
⇒
( A − 6 π r 2 )
⇒
− 12 π r
Now,
( A − 6 π r 2 )
⇒r 2
√
Now,|
| r = √
( − 12 π √
)
Therefore, Volume is maximum atr = √
⇒r 2
6 π r 2
⇒6 π r 2 2 π r 2
⇒4 π r 2
Hence, the volume is maximum if its height is equal to its diameter.
A = 2Ï€rh +
⇒ h =
Now, Volume (V) =
⇒ V =
⇒
⇒
Now,
⇒
Now,
Therefore, Volume is maximum at
⇒
⇒
⇒
Hence, the volume is maximum if its height is equal to its diameter.
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